Applied and Computational Harmonic Analysis最新文献

英文中文

Tikhonov regularization for Gaussian empirical gain maximization in RKHS is consistent RKHS 中高斯经验增益最大化的 Tikhonov 正则化是一致的

IF 2.6 2区数学 Q1 MATHEMATICS, APPLIED

Applied and Computational Harmonic Analysis

Pub Date : 2024-12-09 DOI: 10.1016/j.acha.2024.101735

Yunlong Feng , Qiang Wu

Without imposing light-tailed noise assumptions, we prove that Tikhonov regularization for Gaussian Empirical Gain Maximization (EGM) in a reproducing kernel Hilbert space is consistent and further establish its fast exponential type convergence rates. In the literature, Gaussian EGM was proposed in various contexts to tackle robust estimation problems and has been applied extensively in a great variety of real-world applications. A reproducing kernel Hilbert space is frequently chosen as the hypothesis space, and Tikhonov regularization plays a crucial role in model selection. Although Gaussian EGM has been studied theoretically in a series of papers recently and has been well-understood, theoretical understanding of its Tikhonov regularized variants in RKHS is still limited. Several fundamental challenges remain, especially when light-tailed noise assumptions are absent. To fill the gap and address these challenges, we conduct the present study and make the following contributions. First, under weak moment conditions, we establish a new comparison theorem that enables the investigation of the asymptotic mean calibration properties of regularized Gaussian EGM. Second, under the same weak moment conditions, we show that regularized Gaussian EGM estimators are consistent and further establish their fast exponential-type convergence rates. Our study justifies its feasibility in tackling robust regression problems and explains its robustness from a theoretical viewpoint. Moreover, new technical tools including probabilistic initial upper bounds, confined effective hypothesis spaces, and novel comparison theorems are introduced and developed, which can faciliate the analysis of general regularized empirical gain maximization schemes that fall into the same vein as regularized Gaussian EGM.

在不施加轻尾噪声假设的情况下，证明了再现核Hilbert空间中高斯经验增益最大化（EGM）的Tikhonov正则化是一致的，并进一步建立了其快速指数型收敛速率。在文献中，高斯EGM在各种情况下被提出来解决鲁棒估计问题，并已广泛应用于各种实际应用中。假设空间通常选择再现核希尔伯特空间，吉洪诺夫正则化在模型选择中起着至关重要的作用。虽然最近有一系列论文从理论上对高斯EGM进行了研究，并得到了很好的理解，但对其在RKHS中的Tikhonov正则化变体的理论认识仍然有限。几个基本的挑战仍然存在，特别是在没有光尾噪声假设的情况下。为了填补空白和应对这些挑战，我们进行了本研究，并做出了以下贡献。首先，在弱矩条件下，我们建立了一个新的比较定理，用于研究正则化高斯方程的渐近均值校准性质。其次，在相同的弱矩条件下，我们证明了正则化高斯EGM估计是一致的，并进一步建立了它们的快速指数型收敛速率。我们的研究证明了它在解决鲁棒回归问题上的可行性，并从理论角度解释了它的鲁棒性。此外，引入和发展了新的技术工具，包括概率初始上界、有限有效假设空间和新的比较定理，这些工具可以促进与正则化高斯EGM相同的一般正则化经验增益最大化方案的分析。

{"title":"Tikhonov regularization for Gaussian empirical gain maximization in RKHS is consistent","authors":"Yunlong Feng , Qiang Wu","doi":"10.1016/j.acha.2024.101735","DOIUrl":"10.1016/j.acha.2024.101735","url":null,"abstract":"<div><div>Without imposing light-tailed noise assumptions, we prove that Tikhonov regularization for Gaussian Empirical Gain Maximization (EGM) in a reproducing kernel Hilbert space is consistent and further establish its fast exponential type convergence rates. In the literature, Gaussian EGM was proposed in various contexts to tackle robust estimation problems and has been applied extensively in a great variety of real-world applications. A reproducing kernel Hilbert space is frequently chosen as the hypothesis space, and Tikhonov regularization plays a crucial role in model selection. Although Gaussian EGM has been studied theoretically in a series of papers recently and has been well-understood, theoretical understanding of its Tikhonov regularized variants in RKHS is still limited. Several fundamental challenges remain, especially when light-tailed noise assumptions are absent. To fill the gap and address these challenges, we conduct the present study and make the following contributions. First, under weak moment conditions, we establish a new comparison theorem that enables the investigation of the asymptotic mean calibration properties of regularized Gaussian EGM. Second, under the same weak moment conditions, we show that regularized Gaussian EGM estimators are consistent and further establish their fast exponential-type convergence rates. Our study justifies its feasibility in tackling robust regression problems and explains its robustness from a theoretical viewpoint. Moreover, new technical tools including probabilistic initial upper bounds, confined effective hypothesis spaces, and novel comparison theorems are introduced and developed, which can faciliate the analysis of general regularized empirical gain maximization schemes that fall into the same vein as regularized Gaussian EGM.</div></div>","PeriodicalId":55504,"journal":{"name":"Applied and Computational Harmonic Analysis","volume":"76 ","pages":"Article 101735"},"PeriodicalIF":2.6,"publicationDate":"2024-12-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142823231","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

引用次数: 0

Injectivity of ReLU networks: Perspectives from statistical physics ReLU网络的注入性：来自统计物理学的观点

IF 2.6 2区数学 Q1 MATHEMATICS, APPLIED

Applied and Computational Harmonic Analysis

Pub Date : 2024-12-03 DOI: 10.1016/j.acha.2024.101736

Antoine Maillard , Afonso S. Bandeira , David Belius , Ivan Dokmanić , Shuta Nakajima

When can the input of a ReLU neural network be inferred from its output? In other words, when is the network injective? We consider a single layer,

x \mapsto ReLU (W x)

, with a random Gaussian

m \times n

matrix W, in a high-dimensional setting where

n, m \to \infty

. Recent work connects this problem to spherical integral geometry giving rise to a conjectured sharp injectivity threshold for

α = m / n

by studying the expected Euler characteristic of a certain random set. We adopt a different perspective and show that injectivity is equivalent to a property of the ground state of the spherical perceptron, an important spin glass model in statistical physics. By leveraging the (non-rigorous) replica symmetry-breaking theory, we derive analytical equations for the threshold whose solution is at odds with that from the Euler characteristic. Furthermore, we use Gordon's min–max theorem to prove that a replica-symmetric upper bound refutes the Euler characteristic prediction. Along the way we aim to give a tutorial-style introduction to key ideas from statistical physics in an effort to make the exposition accessible to a broad audience. Our analysis establishes a connection between spin glasses and integral geometry but leaves open the problem of explaining the discrepancies.

何时可以从ReLU神经网络的输出推断其输入？换句话说，什么时候网络是注入的？我们考虑在n，m→∞的高维环境下，具有随机高斯m×n矩阵W的单层x， ReLU（Wx）。最近的工作将这一问题与球面积分几何联系起来，通过研究某随机集的期望欧拉特性，提出了α=m/n的猜想尖锐注入阈值。我们采用了不同的视角，并证明了注入性相当于球形感知器基态的一个性质，球面感知器是统计物理中一个重要的自旋玻璃模型。通过利用（非严格的）副本对称破缺理论，我们推导出阈值的解析方程，其解与欧拉特性的解不一致。此外，我们用Gordon的最小-最大定理证明了一个复制对称上界驳斥了欧拉特征预测。在此过程中，我们的目标是对统计物理学的关键思想进行教程式的介绍，以使广泛的受众能够访问该博览会。我们的分析建立了自旋玻璃和积分几何之间的联系，但没有解释这些差异的问题。

{"title":"Injectivity of ReLU networks: Perspectives from statistical physics","authors":"Antoine Maillard , Afonso S. Bandeira , David Belius , Ivan Dokmanić , Shuta Nakajima","doi":"10.1016/j.acha.2024.101736","DOIUrl":"10.1016/j.acha.2024.101736","url":null,"abstract":"<div><div>When can the input of a ReLU neural network be inferred from its output? In other words, when is the network injective? We consider a single layer, <span><math><mi>x</mi><mo>↦</mo><mrow><mi>ReLU</mi></mrow><mo>(</mo><mi>W</mi><mi>x</mi><mo>)</mo></math></span>, with a random Gaussian <span><math><mi>m</mi><mo>×</mo><mi>n</mi></math></span> matrix <em>W</em>, in a high-dimensional setting where <span><math><mi>n</mi><mo>,</mo><mi>m</mi><mo>→</mo><mo>∞</mo></math></span>. Recent work connects this problem to spherical integral geometry giving rise to a conjectured sharp injectivity threshold for <span><math><mi>α</mi><mo>=</mo><mi>m</mi><mo>/</mo><mi>n</mi></math></span> by studying the expected Euler characteristic of a certain random set. We adopt a different perspective and show that injectivity is equivalent to a property of the ground state of the spherical perceptron, an important spin glass model in statistical physics. By leveraging the (non-rigorous) replica symmetry-breaking theory, we derive analytical equations for the threshold whose solution is at odds with that from the Euler characteristic. Furthermore, we use Gordon's min–max theorem to prove that a replica-symmetric upper bound refutes the Euler characteristic prediction. Along the way we aim to give a tutorial-style introduction to key ideas from statistical physics in an effort to make the exposition accessible to a broad audience. Our analysis establishes a connection between spin glasses and integral geometry but leaves open the problem of explaining the discrepancies.</div></div>","PeriodicalId":55504,"journal":{"name":"Applied and Computational Harmonic Analysis","volume":"76 ","pages":"Article 101736"},"PeriodicalIF":2.6,"publicationDate":"2024-12-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142790150","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

引用次数: 0

Group projected subspace pursuit for block sparse signal reconstruction: Convergence analysis and applications 1 块稀疏信号重构中的群投影子空间追踪：收敛性分析及应用

IF 2.6 2区数学 Q1 MATHEMATICS, APPLIED

Applied and Computational Harmonic Analysis

Pub Date : 2024-11-28 DOI: 10.1016/j.acha.2024.101726

Roy Y. He , Haixia Liu , Hao Liu

In this paper, we present a convergence analysis of the Group Projected Subspace Pursuit (GPSP) algorithm proposed by He et al. [26] (Group Projected subspace pursuit for IDENTification of variable coefficient differential equations (GP-IDENT), Journal of Computational Physics, 494, 112526) and extend its application to general tasks of block sparse signal recovery. Given an observation y and sampling matrix A, we focus on minimizing the approximation error

{‖ A c - y ‖}_{2}^{2}

with respect to the signal c with block sparsity constraints. We prove that when the sampling matrix A satisfies the Block Restricted Isometry Property (BRIP) with a sufficiently small Block Restricted Isometry Constant (BRIC), GPSP exactly recovers the true block sparse signals. When the observations are noisy, this convergence property of GPSP remains valid if the magnitude of the true signal is sufficiently large. GPSP selects the features by subspace projection criterion (SPC) for candidate inclusion and response magnitude criterion (RMC) for candidate exclusion. We compare these criteria with counterparts of other state-of-the-art greedy algorithms. Our theoretical analysis and numerical ablation studies reveal that SPC is critical to the superior performances of GPSP, and that RMC can enhance the robustness of feature identification when observations contain noises. We test and compare GPSP with other methods in diverse settings, including heterogeneous random block matrices, inexact observations, face recognition, and PDE identification. We find that GPSP outperforms the other algorithms in most cases for various levels of block sparsity and block sizes, justifying its effectiveness for general applications.

本文对He et al. [26] （Group Projected Subspace Pursuit for IDENTification of variable coefficient differential equations (GP-IDENT), Journal of Computational Physics, 494, 112526）提出的Group Projected Subspace Pursuit （GPSP）算法进行了收敛性分析，并将其应用于块稀疏信号恢复的一般任务。给定观测值y和采样矩阵A，我们专注于最小化相对于具有块稀疏性约束的信号c的近似误差‖Ac−y‖22。证明了当采样矩阵A满足块受限等距特性（BRIP）且块受限等距常数（BRIC）足够小时，GPSP能准确地恢复出真实的块稀疏信号。当观测值有噪声时，如果真信号的幅度足够大，GPSP的这种收敛性仍然有效。GPSP通过子空间投影准则（SPC）选择候选包含特征，通过响应大小准则（RMC）选择候选排除特征。我们将这些标准与其他最先进的贪婪算法进行比较。我们的理论分析和数值消融研究表明，SPC是GPSP优越性能的关键，RMC可以增强观测值包含噪声时特征识别的鲁棒性。我们在不同的环境下测试并比较了GPSP和其他方法，包括异构随机块矩阵、不精确观察、人脸识别和PDE识别。我们发现，在大多数情况下，对于不同级别的块稀疏性和块大小，GPSP优于其他算法，证明其在一般应用中的有效性。

{"title":"Group projected subspace pursuit for block sparse signal reconstruction: Convergence analysis and applications 1","authors":"Roy Y. He , Haixia Liu , Hao Liu","doi":"10.1016/j.acha.2024.101726","DOIUrl":"10.1016/j.acha.2024.101726","url":null,"abstract":"<div><div>In this paper, we present a convergence analysis of the Group Projected Subspace Pursuit (GPSP) algorithm proposed by He et al. <span><span>[26]</span></span> (Group Projected subspace pursuit for IDENTification of variable coefficient differential equations (GP-IDENT), <em>Journal of Computational Physics</em>, 494, 112526) and extend its application to general tasks of block sparse signal recovery. Given an observation <strong>y</strong> and sampling matrix <strong>A</strong>, we focus on minimizing the approximation error <span><math><msubsup><mrow><mo>‖</mo><mi>A</mi><mi>c</mi><mo>−</mo><mi>y</mi><mo>‖</mo></mrow><mrow><mn>2</mn></mrow><mrow><mn>2</mn></mrow></msubsup></math></span> with respect to the signal <strong>c</strong> with block sparsity constraints. We prove that when the sampling matrix <strong>A</strong> satisfies the Block Restricted Isometry Property (BRIP) with a sufficiently small Block Restricted Isometry Constant (BRIC), GPSP exactly recovers the true block sparse signals. When the observations are noisy, this convergence property of GPSP remains valid if the magnitude of the true signal is sufficiently large. GPSP selects the features by subspace projection criterion (SPC) for candidate inclusion and response magnitude criterion (RMC) for candidate exclusion. We compare these criteria with counterparts of other state-of-the-art greedy algorithms. Our theoretical analysis and numerical ablation studies reveal that SPC is critical to the superior performances of GPSP, and that RMC can enhance the robustness of feature identification when observations contain noises. We test and compare GPSP with other methods in diverse settings, including heterogeneous random block matrices, inexact observations, face recognition, and PDE identification. We find that GPSP outperforms the other algorithms in most cases for various levels of block sparsity and block sizes, justifying its effectiveness for general applications.</div></div>","PeriodicalId":55504,"journal":{"name":"Applied and Computational Harmonic Analysis","volume":"75 ","pages":"Article 101726"},"PeriodicalIF":2.6,"publicationDate":"2024-11-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142759319","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

引用次数: 0

Anisotropic refinable functions and the tile B-splines 各向异性可细化函数和平铺b样条

IF 2.6 2区数学 Q1 MATHEMATICS, APPLIED

Applied and Computational Harmonic Analysis

Pub Date : 2024-11-28 DOI: 10.1016/j.acha.2024.101727

Vladimir Yu. Protasov , Tatyana Zaitseva

The regularity of refinable functions has been analysed in an extensive literature and is well-understood in two cases: 1) univariate 2) multivariate with an isotropic dilation matrix. The general (non-isotropic) case offered a great resistance. It was not before 2019 that the non-isotropic case was done by developing the matrix method. In this paper we make the next step and extend the Littlewood-Paley type method, which is very efficient in the aforementioned special cases, to general equations with arbitrary dilation matrices. This gives formulas for the higher order regularity in

W_{2}^{k} (R^{n})

by means of the Perron eigenvalue of a finite-dimensional linear operator on a special cone. Applying those results to recently introduced tile B-splines, we prove that they can have a higher smoothness than the classical ones of the same order. Moreover, the two-digit tile B-splines have the minimal support of the mask among all refinable functions of the same order of approximation. This proves, in particular, the lowest algorithmic complexity of the corresponding subdivision schemes. Examples and numerical results are provided.

可细化函数的正则性在两种情况下得到了很好的理解：1)单变量2)具有各向同性膨胀矩阵的多变量。一般（非各向同性）情况提供了很大的阻力。直到2019年，才通过开发矩阵方法完成了非各向同性的情况。本文进一步将在上述特殊情况下非常有效的Littlewood-Paley型方法推广到具有任意膨胀矩阵的一般方程。利用特殊锥上有限维线性算子的Perron特征值，给出了W2k（Rn）中高阶正则性的表达式。将这些结果应用到最近引入的平铺b样条上，证明了它们比经典的同阶平铺b样条具有更高的平滑性。此外，两位数平铺b样条在所有同阶近似的可细化函数中具有最小的掩模支持。这特别证明了相应细分方案的算法复杂度最低。给出了算例和数值结果。

{"title":"Anisotropic refinable functions and the tile B-splines","authors":"Vladimir Yu. Protasov , Tatyana Zaitseva","doi":"10.1016/j.acha.2024.101727","DOIUrl":"10.1016/j.acha.2024.101727","url":null,"abstract":"<div><div>The regularity of refinable functions has been analysed in an extensive literature and is well-understood in two cases: 1) univariate 2) multivariate with an isotropic dilation matrix. The general (non-isotropic) case offered a great resistance. It was not before 2019 that the non-isotropic case was done by developing the matrix method. In this paper we make the next step and extend the Littlewood-Paley type method, which is very efficient in the aforementioned special cases, to general equations with arbitrary dilation matrices. This gives formulas for the higher order regularity in <span><math><msubsup><mrow><mi>W</mi></mrow><mrow><mn>2</mn></mrow><mrow><mi>k</mi></mrow></msubsup><mo>(</mo><msup><mrow><mi>R</mi></mrow><mrow><mi>n</mi></mrow></msup><mo>)</mo></math></span> by means of the Perron eigenvalue of a finite-dimensional linear operator on a special cone. Applying those results to recently introduced tile B-splines, we prove that they can have a higher smoothness than the classical ones of the same order. Moreover, the two-digit tile B-splines have the minimal support of the mask among all refinable functions of the same order of approximation. This proves, in particular, the lowest algorithmic complexity of the corresponding subdivision schemes. Examples and numerical results are provided.</div></div>","PeriodicalId":55504,"journal":{"name":"Applied and Computational Harmonic Analysis","volume":"75 ","pages":"Article 101727"},"PeriodicalIF":2.6,"publicationDate":"2024-11-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142759438","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

引用次数: 0

Scale dependencies and self-similar models with wavelet scattering spectra 小波散射谱的尺度依赖性和自相似模型

IF 2.6 2区数学 Q1 MATHEMATICS, APPLIED

Applied and Computational Harmonic Analysis

Pub Date : 2024-11-19 DOI: 10.1016/j.acha.2024.101724

Rudy Morel , Gaspar Rochette , Roberto Leonarduzzi , Jean-Philippe Bouchaud , Stéphane Mallat

Multi-scale non-Gaussian time-series having stationary increments appear in a wide range of applications, particularly in finance and physics. We introduce stochastic models that capture intermittency phenomena such as crises or bursts of activity, time reversal asymmetries, and that can be estimated from a single realization of size N. Variations at multiple scales are separated with a wavelet transform. Non-Gaussian properties appear through dependencies of wavelet coefficients across scales. We define maximum entropy models from the joint correlation across time and scales of wavelet coefficients and their modulus. Diagonal matrix approximations are estimated with a wavelet representation of this joint correlation. The resulting diagonals define

O (\log^{3} N)

moments that are called scattering spectra. A notion of wide-sense self-similarity is defined from the invariance of scattering spectra to scaling, which can be tested numerically on a single realization. We study the accuracy of maximum entropy scattering spectra models for fractional Brownian motions, Hawkes processes, multifractal random walks, as well as financial and turbulent time-series.

具有静态增量的多尺度非高斯时间序列出现在广泛的应用中，尤其是在金融和物理领域。我们引入的随机模型能捕捉间歇现象，如危机或活动爆发、时间反转不对称，并能从大小为 N 的单一实现中进行估算。非高斯特性通过跨尺度小波系数的依赖性显现出来。我们根据小波系数及其模量在时间和尺度上的联合相关性定义最大熵模型。对角矩阵近似值是用这种联合相关性的小波表示来估算的。由此得到的对角矩阵定义了 O(log3N) 矩，称为散射谱。根据散射谱对缩放的不变性，我们定义了广义自相似性的概念，并可在单个实现上对其进行数值测试。我们研究了分数布朗运动、霍克斯过程、多分形随机游走以及金融和湍流时间序列的最大熵散射谱模型的准确性。

{"title":"Scale dependencies and self-similar models with wavelet scattering spectra","authors":"Rudy Morel , Gaspar Rochette , Roberto Leonarduzzi , Jean-Philippe Bouchaud , Stéphane Mallat","doi":"10.1016/j.acha.2024.101724","DOIUrl":"10.1016/j.acha.2024.101724","url":null,"abstract":"<div><div>Multi-scale non-Gaussian time-series having stationary increments appear in a wide range of applications, particularly in finance and physics. We introduce stochastic models that capture intermittency phenomena such as crises or bursts of activity, time reversal asymmetries, and that can be estimated from a single realization of size <em>N</em>. Variations at multiple scales are separated with a wavelet transform. Non-Gaussian properties appear through dependencies of wavelet coefficients across scales. We define maximum entropy models from the joint correlation across time and scales of wavelet coefficients and their modulus. Diagonal matrix approximations are estimated with a wavelet representation of this joint correlation. The resulting diagonals define <span><math><mi>O</mi><mo>(</mo><msup><mrow><mi>log</mi></mrow><mrow><mn>3</mn></mrow></msup><mo>⁡</mo><mi>N</mi><mo>)</mo></math></span> moments that are called scattering spectra. A notion of wide-sense self-similarity is defined from the invariance of scattering spectra to scaling, which can be tested numerically on a single realization. We study the accuracy of maximum entropy scattering spectra models for fractional Brownian motions, Hawkes processes, multifractal random walks, as well as financial and turbulent time-series.</div></div>","PeriodicalId":55504,"journal":{"name":"Applied and Computational Harmonic Analysis","volume":"75 ","pages":"Article 101724"},"PeriodicalIF":2.6,"publicationDate":"2024-11-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142696456","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

引用次数: 0

Multidimensional unstructured sparse recovery via eigenmatrix 通过特征矩阵进行多维非结构化稀疏恢复

IF 2.6 2区数学 Q1 MATHEMATICS, APPLIED

Applied and Computational Harmonic Analysis

Pub Date : 2024-11-19 DOI: 10.1016/j.acha.2024.101725

Lexing Ying

This note considers the multidimensional unstructured sparse recovery problems. Examples include Fourier inversion and sparse deconvolution. The eigenmatrix is a data-driven construction with desired approximate eigenvalues and eigenvectors proposed for the one-dimensional problems. This note extends the eigenmatrix approach to multidimensional problems, providing a rather unified treatment for general kernels and unstructured sampling grids in both real and complex settings. Numerical results are provided to demonstrate the performance of the proposed method.

本说明探讨了多维非结构稀疏恢复问题。例如傅立叶反演和稀疏解卷积。特征矩阵是一种数据驱动的构造，针对一维问题提出了所需的近似特征值和特征向量。本说明将特征矩阵方法扩展到多维问题，为真实和复杂环境中的一般核和非结构化采样网格提供了相当统一的处理方法。本文提供了数值结果，以证明所提方法的性能。

引用次数: 0

The beltway problem over orthogonal groups 正交群上的带路问题

IF 2.6 2区数学 Q1 MATHEMATICS, APPLIED

Applied and Computational Harmonic Analysis

Pub Date : 2024-11-15 DOI: 10.1016/j.acha.2024.101723

Tamir Bendory, Dan Edidin, Oscar Mickelin

The classical beltway problem entails recovering a set of points from their unordered pairwise distances on the circle. This problem can be viewed as a special case of the crystallographic phase retrieval problem of recovering a sparse signal from its periodic autocorrelation. Based on this interpretation, and motivated by cryo-electron microscopy, we suggest a natural generalization to orthogonal groups: recovering a sparse signal, up to an orthogonal transformation, from its autocorrelation over the orthogonal group. If the support of the signal is collision-free, we bound the number of solutions to the beltway problem over orthogonal groups, and prove that this bound is exactly one when the support of the signal is radially collision-free (i.e., the support points have distinct magnitudes). We also prove that if the pairwise products of the signal's weights are distinct, then the autocorrelation determines the signal uniquely, up to an orthogonal transformation. We conclude the paper by considering binary signals and show that in this case, the collision-free condition need not be sufficient to determine signals up to orthogonal transformation.

经典的带状线问题需要从圆周上无序的成对距离中恢复一组点。这个问题可以看作是晶体学相位检索问题的一个特例，即从周期性自相关中恢复稀疏信号。基于这一解释，并受冷冻电子显微镜的启发，我们提出了对正交群的自然概括：从正交群上的自相关中恢复稀疏信号，直至正交变换。如果信号的支撑点是无碰撞的，我们将对正交群上的带路问题解的数量进行约束，并证明当信号的支撑点是径向无碰撞的（即支撑点具有不同的大小）时，这个约束正好是一。我们还证明，如果信号权重的成对乘积是不同的，那么自相关决定了信号的唯一性，直到正交变换为止。最后，我们考虑了二进制信号，并证明在这种情况下，无碰撞条件不一定足以决定信号的正交变换。

{"title":"The beltway problem over orthogonal groups","authors":"Tamir Bendory, Dan Edidin, Oscar Mickelin","doi":"10.1016/j.acha.2024.101723","DOIUrl":"10.1016/j.acha.2024.101723","url":null,"abstract":"<div><div>The classical beltway problem entails recovering a set of points from their unordered pairwise distances on the circle. This problem can be viewed as a special case of the crystallographic phase retrieval problem of recovering a sparse signal from its periodic autocorrelation. Based on this interpretation, and motivated by cryo-electron microscopy, we suggest a natural generalization to orthogonal groups: recovering a sparse signal, up to an orthogonal transformation, from its autocorrelation over the orthogonal group. If the support of the signal is collision-free, we bound the number of solutions to the beltway problem over orthogonal groups, and prove that this bound is exactly one when the support of the signal is radially collision-free (i.e., the support points have distinct magnitudes). We also prove that if the pairwise products of the signal's weights are distinct, then the autocorrelation determines the signal uniquely, up to an orthogonal transformation. We conclude the paper by considering binary signals and show that in this case, the collision-free condition need not be sufficient to determine signals up to orthogonal transformation.</div></div>","PeriodicalId":55504,"journal":{"name":"Applied and Computational Harmonic Analysis","volume":"74 ","pages":"Article 101723"},"PeriodicalIF":2.6,"publicationDate":"2024-11-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142696470","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

引用次数: 0

On quadrature for singular integral operators with complex symmetric quadratic forms 关于具有复对称二次形式的奇异积分算子的正交性

IF 2.6 2区数学 Q1 MATHEMATICS, APPLIED

Applied and Computational Harmonic Analysis

Pub Date : 2024-11-13 DOI: 10.1016/j.acha.2024.101721

Jeremy Hoskins , Manas Rachh , Bowei Wu

This paper describes a trapezoidal quadrature method for the discretization of weakly singular, and hypersingular boundary integral operators with complex symmetric quadratic forms. Such integral operators naturally arise when complex coordinate methods or complexified contour methods are used for the solution of time-harmonic acoustic and electromagnetic interface problems in three dimensions. The quadrature is an extension of a locally corrected punctured trapezoidal rule in parameter space wherein the correction weights are determined by fitting moments of error in the punctured trapezoidal rule, which is known analytically in terms of the Epstein zeta function. In this work, we analyze the analytic continuation of the Epstein zeta function and the generalized Wigner limits to complex quadratic forms; this analysis is essential to apply the fitting procedure for computing the correction weights. We illustrate the high-order convergence of this approach through several numerical examples.

本文介绍了一种梯形正交方法，用于离散化具有复对称二次方形式的弱奇异和超奇异边界积分算子。当使用复坐标法或复等值线法求解三维时谐声学和电磁界面问题时，自然会出现此类积分算子。正交是局部修正的点阵梯形法则在参数空间中的扩展，其中修正权重由点阵梯形法则中的误差拟合矩决定，而误差拟合矩是通过爱泼斯坦兹塔函数解析得知的。在这项工作中，我们分析了爱泼斯坦zeta函数的解析延续和广义维格纳极限的复二次型；这一分析对于应用拟合程序计算修正权重至关重要。我们通过几个数值示例说明了这种方法的高阶收敛性。

引用次数: 0

Gaussian approximation for the moving averaged modulus wavelet transform and its variants 移动平均模小波变换的高斯近似及其变体

IF 2.6 2区数学 Q1 MATHEMATICS, APPLIED

Applied and Computational Harmonic Analysis

Pub Date : 2024-11-13 DOI: 10.1016/j.acha.2024.101722

Gi-Ren Liu , Yuan-Chung Sheu , Hau-Tieng Wu

The moving average of the complex modulus of the analytic wavelet transform provides a robust time-scale representation for signals to small time shifts and deformation. In this work, we derive the Wiener chaos expansion of this representation for stationary Gaussian processes by the Malliavin calculus and combinatorial techniques. The expansion allows us to obtain a lower bound for the Wasserstein distance between the time-scale representations of two long-range dependent Gaussian processes in terms of Hurst indices. Moreover, we apply the expansion to establish an upper bound for the smooth Wasserstein distance and the Kolmogorov distance between the distributions of a random vector derived from the time-scale representation and its normal counterpart. It is worth mentioning that the expansion consists of infinite Wiener chaos, and the projection coefficients converge to zero slowly as the order of the Wiener chaos increases. We provide a rational-decay upper bound for these distribution distances, the rate of which depends on the nonlinear transformation of the amplitude of the complex wavelet coefficients.

解析小波变换复数模的移动平均值为信号提供了一种稳健的时间尺度表示法，可用于较小的时间偏移和变形。在这项工作中，我们通过马利亚文微积分和组合技术，为静态高斯过程推导出了这一表示的维纳混沌扩展。通过该扩展，我们获得了两个长程依赖高斯过程的时间尺度表示之间以赫斯特指数为单位的瓦瑟斯坦距离下限。此外，我们还应用扩展建立了平滑瓦瑟斯坦距离的上界，以及由时间尺度表示得出的随机向量的分布与其正态对应物之间的科尔莫哥洛夫距离的上界。值得一提的是，扩展由无限维纳混沌组成，随着维纳混沌阶数的增加，投影系数会慢慢趋近于零。我们提供了这些分布距离的有理衰减上限，其速率取决于复小波系数振幅的非线性变换。

{"title":"Gaussian approximation for the moving averaged modulus wavelet transform and its variants","authors":"Gi-Ren Liu , Yuan-Chung Sheu , Hau-Tieng Wu","doi":"10.1016/j.acha.2024.101722","DOIUrl":"10.1016/j.acha.2024.101722","url":null,"abstract":"<div><div>The moving average of the complex modulus of the analytic wavelet transform provides a robust time-scale representation for signals to small time shifts and deformation. In this work, we derive the Wiener chaos expansion of this representation for stationary Gaussian processes by the Malliavin calculus and combinatorial techniques. The expansion allows us to obtain a lower bound for the Wasserstein distance between the time-scale representations of two long-range dependent Gaussian processes in terms of Hurst indices. Moreover, we apply the expansion to establish an upper bound for the smooth Wasserstein distance and the Kolmogorov distance between the distributions of a random vector derived from the time-scale representation and its normal counterpart. It is worth mentioning that the expansion consists of infinite Wiener chaos, and the projection coefficients converge to zero slowly as the order of the Wiener chaos increases. We provide a rational-decay upper bound for these distribution distances, the rate of which depends on the nonlinear transformation of the amplitude of the complex wavelet coefficients.</div></div>","PeriodicalId":55504,"journal":{"name":"Applied and Computational Harmonic Analysis","volume":"74 ","pages":"Article 101722"},"PeriodicalIF":2.6,"publicationDate":"2024-11-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142658789","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

引用次数: 0

Naimark-spatial families of equichordal tight fusion frames 等弦紧密融合框架的奈马克空间族

IF 2.6 2区数学 Q1 MATHEMATICS, APPLIED

Applied and Computational Harmonic Analysis

Pub Date : 2024-11-08 DOI: 10.1016/j.acha.2024.101720

Matthew Fickus, Benjamin R. Mayo, Cody E. Watson

An equichordal tight fusion frame (

) is a finite sequence of equi-dimensional subspaces of a Euclidean space that achieves equality in Conway, Hardin and Sloane's simplex bound. Every

is a type of optimal Grassmannian code, being a way to arrange a given number of members of a Grassmannian so that the minimal chordal distance between any pair of them is as large as possible. Any nontrivial

has both a Naimark complement and spatial complement which themselves are

s. We show that taking iterated alternating Naimark and spatial complements of any

of at least five subspaces yields an infinite family of

s with pairwise distinct parameters. Generalizing a method by King, we then construct

s from difference families for finite abelian groups, and use our Naimark-spatial theory to gauge their novelty.

等弦密融合框（）是欧几里得空间的等维子空间的有限序列，它在康威、哈丁和斯隆的单数约束中达到相等。每一个都是一种最优格拉斯曼编码，是一种排列给定数量的格拉斯曼成员，使任意一对成员之间的最小弦距尽可能大的方法。我们的研究表明，对至少五个子空间中的任意子空间进行迭代交替的奈马克补集和空间补集，就能得到一个具有成对不同参数的无穷 s 族。然后，我们推广了金的一种方法，从有限无性群的差分族中构造出 s，并利用我们的奈马克空间理论来衡量它们的新颖性。

引用次数: 0

首页上一页

下一页尾页

类型

全部化学•材料生命科学医学物理工程技术环境•农林材料科学地球科学法学管理学化学环境科学与生态学计算机科学教育学经济学农林科学人文科学生物学数学物理与天体物理心理学综合性期刊其他工业工程理学历史学农学文学信息工程

数据库

全部 ACS Publications Elsevier ieeexplore Springer The Royal Society of Chemistry Wiley

期刊

Applied and Computational Harmonic Analysis

全部 Acc. Chem. Res. ACS Applied Bio Materials ACS Appl. Electron. Mater. ACS Appl. Energy Mater. ACS Appl. Mater. Interfaces ACS Appl. Nano Mater. ACS Appl. Polym. Mater. ACS BIOMATER-SCI ENG ACS Catal. ACS Cent. Sci. ACS Chem. Biol. ACS Chemical Health & Safety ACS Chem. Neurosci. ACS Comb. Sci. ACS Earth Space Chem. ACS Energy Lett. ACS Infect. Dis. ACS Macro Lett. ACS Mater. Lett. ACS Med. Chem. Lett. ACS Nano ACS Omega ACS Photonics ACS Sens. ACS Sustainable Chem. Eng. ACS Synth. Biol. Anal. Chem. BIOCHEMISTRY-US Bioconjugate Chem. BIOMACROMOLECULES Chem. Res. Toxicol. Chem. Rev. Chem. Mater. CRYST GROWTH DES ENERG FUEL Environ. Sci. Technol. Environ. Sci. Technol. Lett. Eur. J. Inorg. Chem. IND ENG CHEM RES Inorg. Chem. J. Agric. Food. Chem. J. Chem. Eng. Data J. Chem. Educ. J. Chem. Inf. Model. J. Chem. Theory Comput. J. Med. Chem. J. Nat. Prod. J PROTEOME RES J. Am. Chem. Soc. LANGMUIR MACROMOLECULES Mol. Pharmaceutics Nano Lett. Org. Lett. ORG PROCESS RES DEV ORGANOMETALLICS J. Org. Chem. J. Phys. Chem. J. Phys. Chem. A J. Phys. Chem. B J. Phys. Chem. C J. Phys. Chem. Lett. Analyst Anal. Methods Biomater. Sci. Catal. Sci. Technol. Chem. Commun. Chem. Soc. Rev. CHEM EDUC RES PRACT CRYSTENGCOMM Dalton Trans. Energy Environ. Sci. ENVIRON SCI-NANO ENVIRON SCI-PROC IMP ENVIRON SCI-WAT RES Faraday Discuss. Food Funct. Green Chem. Inorg. Chem. Front. Integr. Biol. J. Anal. At. Spectrom. J. Mater. Chem. A J. Mater. Chem. B J. Mater. Chem. C Lab Chip Mater. Chem. Front. Mater. Horiz. MEDCHEMCOMM Metallomics Mol. Biosyst. Mol. Syst. Des. Eng. Nanoscale Nanoscale Horiz. Nat. Prod. Rep. New J. Chem. Org. Biomol. Chem. Org. Chem. Front. PHOTOCH PHOTOBIO SCI PCCP Polym. Chem.

﹀